Max Area of Triangle with Two Moving Points
2020-05-12 12:33
This applet demonstrates constructing a triangle with two moving points in plane geometry to explore its maximum area. By dynamically adjusting the points, it visually shows area variations and the corresponding function graph. Combining polygons, segments, vectors, and loci, it helps understand the connection between geometric optimization and function graphs.
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.579.0" app="classic" platform="d" id="4ba8bceb-5fba-425f-8853-761bcd91e41e" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1920" height="1030" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.13020833333333334" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="330" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1636" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="250" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1636" height="776"/>
<coordSystem xZero="912.1693738467221" yZero="710.2287708215122" scale="50.71180891022334" yscale="50.71180891022339"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="text1" exp=""【2018广东】已知Rt△OAB，∠AOB=90°，∠ABO=30°，
斜边OB=4，将Rt△OAB绕点O顺时针旋转60°，
连接BC．连接BC，点M,N同时从点O出发，
在△OCB边上运动，点M沿路径O-C-B匀速运动，
N沿路径O-B-C匀速运动，当两点相遇时运动停止，
已知点M的运动速度为1.5单位/秒，点N的运动速度
为1单位/秒，设运动时间为x秒，△OMN的面积y为，
求当x为何值时y取得最大值？最大值为多少？""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-12.180000000000001" y="11.859999999999998" z="1.0"/>
</element>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-34" y="26"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-11.425" y="0.52" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="B" exp="A + (0, (2 * sqrt(3)))" type="point" />
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-26" y="4"/>
<labelMode val="0"/>
<coords x="-11.425" y="3.9841016151377544" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="O" exp="A + (2, 0)" type="point" />
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="12" y="26"/>
<labelMode val="0"/>
<coords x="-9.425" y="0.52" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="C" exp="A + (2 * sqrt(3))" type="point" />
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-7.960898384862246" y="3.9841016151377544" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="C" a3="O"/>
<output a0="q1" a1="a_1" a2="b" a3="c" a4="o"/>
</command>
<element type="polygon" label="q1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="51" g="51" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-3.4641016151377544" y="0.0" z="-39.577360952948844"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="3.4641016151377544" z="-13.80133283987163"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.4641016151377544" y="-1.4641016151377544" z="33.41049056254497"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="o">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-10" y="52"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="-2.0" z="1.04"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="PolyLine">
<input a0="O" a1="B" a2="C"/>
<output a0="f"/>
</command>
<element type="polyline" label="f">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-38" y="14"/>
<labelMode val="0"/>
</element>
<command name="PolyLine">
<input a0="O" a1="C" a2="B"/>
<output a0="g"/>
</command>
<element type="polyline" label="g">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="58" y="19"/>
<labelMode val="0"/>
</element>
<element type="numeric" label="d">
<value val="1.8"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="8" absoluteScreenLocation="true" width="200.0" x="122.0" y="462.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<command name="UnitVector">
<input a0="Segment[O, B]"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.5" y="0.8660254037844386" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="O"/>
</element>
<command name="UnitVector">
<input a0="Segment[B, C]"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0" y="-0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="B"/>
</element>
<command name="UnitVector">
<input a0="Segment[C, O]"/>
<output a0="w"/>
</command>
<element type="vector" label="w">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.389306214312518" y="-0.9211083929145667" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="C"/>
</element>
<command name="Segment">
<input a0="O" a1="B"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.4641016151377544" y="-2.0" z="-31.609157722673338"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="T" exp="(h + c + b) / (1.5 + 1)" />
<element type="numeric" label="T">
<value val="4.489959417563267"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
</element>
<element type="numeric" label="e">
<value val="1.66"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="4" y="12"/>
<labelMode val="3"/>
<slider min="0" max="a" absoluteScreenLocation="true" width="200.0" x="325.0" y="369.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<caption val="$x$"/>
</element>
<command name="If">
<input a0="e ≤ h" a1="O + (e * u)" a2="B + ((e - h) * v)"/>
<output a0="N"/>
</command>
<element type="point" label="N">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-26" y="7"/>
<labelMode val="0"/>
<coords x="-10.255" y="1.957602170282168" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="If">
<input a0="e ≤ c / 1.5" a1="O - ((1.5 * e) * w)" a2="C - ((1.5 * (e - c / 1.5)) * v)"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="10" y="16"/>
<labelMode val="0"/>
<coords x="-8.45562752636183" y="2.813559898357271" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="O" a1="M" a2="N"/>
<output a0="t1" a1="n" a2="o_1" a3="m"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="n">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.293559898357271" y="0.9693724736381704" z="-22.12087572830913"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="o_1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.855957728075103" y="-1.7993724736381704" z="12.300301960950254"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="m">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.437602170282168" y="0.8300000000000001" z="13.117800454909435"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="D" exp="(e, t1)" type="point" />
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.66" y="1.6486133437752795" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="a">
<value val="1.66"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="4.49" absoluteScreenLocation="true" width="200.0" x="324.0" y="409.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="SetValue(e,a)"/>
</element>
<command name="Locus">
<input a0="D" a1="e"/>
<output a0="loc1"/>
</command>
<element type="locus" label="loc1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
</construction>
</geogebra>